Prove that \[\lim_{x\to a}\,f(x)=L\] if and only if \[\lim_{x\to a}\,(f(x)-L)=0.\]

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Lesson Parent: 
Function Limits III: The Gory Details
Problem: 

Prove that \[\lim_{x\to a}\,f(x)=L\] if and only if \[\lim_{x\to a}\,(f(x)-L)=0.\]

Answer: 

It is true that \[\lim_{x\to a}\,f(x)=L\] if and only if \[\lim_{x\to a}\,(f(x)-L)=0.\]

SAMPLE SOLUTION

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Solution: 

In general, we can have statements like "A if and only if B" where A and B are sentences that can be either true or false. To prove this we need to show that "if A, then B" and "if B, then A".

Therefore, to prove this we need to prove that:

The proof is complete.